OFFSET
0,3
COMMENTS
Alternatively, these are partitions with a part appearing fewer times than each of the others.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (Product_{j>=1} (1 + x^(j*m)/(1 - x^j))). - Andrew Howroyd, May 04 2023
EXAMPLE
The partition (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (2111) (411) (511) (422)
(11111) (3111) (2221) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (22211)
(211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]==1&]], {n, 0, 30}]
PROG
(PARI) seq(n) = my(A=O(x*x^n)); Vec(sum(m=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 30 2023
STATUS
approved